2 Answers
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Welders have one main purpose: Repairing stuff. They can be used on structures as well as directly on Marines (which repairs their armor, but not their health).

When a welder is used on an unbuilt structure, it builds it (like when holding your Use key) instead of repairing it. Based on my tests, welders build structures 10% faster than the standard build tool, but this is only a minor advantage when you consider that most buildings only take 10-20 seconds to build.

However, using a welder on a destroyed power node (not just an unbuilt one) will repair it over twice as fast as the standard build tool (from 32 seconds down to 15), saving precious time in what is undoubtedly an unpleasant situation.

Since the welder replaces the axe, you can also use it as a melee weapon. While it has a slightly higher base damage compared to the axe, it lacks the axe's 2x structure damage bonus, making it a poorer choice when you're taking down buildings. (Except for Cysts and Clogs, which are flammable and take 3x damage from the welder's flame.)

(Historical note: In NS1, welders could also be used to permanently open/close certain doors on some maps. This feature didn't make it into NS2, but it's something the developers want to do.)